We will also need the continuity bound for reflected entropy [1]}, which follows from applying the Fannes-Audenaert inequality [2]}, [3]} to the reflected entropy. In fact we will need a Rényi generalization of this inequality, which is usually considered to be a weak bound [4]}:
\(\left|S_n( \rho _{AA^\star }^{(1)} ) - S_n( \rho _{AA^\star }^{(m)} ) \right| &\le \frac{ \chi _A^{2 (n-1)}}{n-1} \left(1-(1- T)^n -(\chi _A^2-1)^{1-n} T^n \right)\, \qquad n \ge 1 \\& \le \frac{ n\chi _A^{2 (n-1)}}{n-1} T\) 
